Random Graph-Homomorphisms and Logarithmic Degree

A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph G to the infinite line Z. It is shown that if the maximal degree of G is `sub-logarithmic', then the range of such a homomorphism is super-constant. Furthermore, some examples are provided, suggesting that perhaps for graphs with super-logarithmic degree, the range of a typical homomorphism is bounded. In particular, a sharp transition is shown for a specific family of graphs C_{n,k} (which is the tensor product of the n-cycle and a complete graph, with self-loops, of size k). That is, given any function psi(n) tending to infinity, the range of a typical homomorphism of C_{n,k} is super-constant for k = 2 log(n) - psi(n), and is 3 for k = 2 log(n) + psi(n).


Introduction
A graph homomorphism from a graph G to a graph H is a map from the vertex set of G to the vertex set of H, that maps edges to edges. By a homomorphism of G we mean a graph homomorphism from G to the infinite line Z. Thus, a homomorphism of G maps adjacent vertices to adjacent integers. We note that the uniform measure on the set of all homomorphisms of G, that send some fixed vertex to 0, generalizes the concept of random walks on Z. Indeed, a random homomorphism of the k-line is a random walk of length k on Z.
So, random homomorphisms of a graph G, are also referred to as G-indexed random walks.
Tree-indexed random walks were studied by Benjamini and Peres in [2]. For results concerning random homomorphisms of general graphs see [1,8]. [3] deals with connections between random homomorphisms and the Gaussian random field. For other related 2-dimensional height models in physics see [7,9].
A key quantity for our first result, Theorem 2.1, is V (r), the maximal size of a ball of radius r in G. Theorem 2.1 states that for every r such that V (r) is at most 1 2 log(|G|), the range of a random homomorphism is greater than r, with high probability. If d is the maximal degree in G, then V (r) is at most (d + 1) r . Thus, Theorem 2.1 implies that for graphs of 'small enough' degree, the range of a homomorphism is typically 'large' (see Corollary 2.2).
We stress that this is only a sufficient condition for large range, and not a necessary one. For example, consider the log(n)-regular tree of size n. Already for r = 1, a ball of radius r has at least log(n) vertices, so the assertion of Theorem 2.1 is trivial. However, the range of a typical homomorphism of this tree is of size at least Ω(log(n)/ log log(n)).
The next natural question is: How tight is this lower bound? That is, are there examples of graphs of logarithmic degree that have bounded range (as the size of the graph grows to infinity)? This can be decided via a result of Kahn [6]. Kahn's results states that there exists a constant b ∈ N, such that the range of a random homomorphism of Q d , the discrete cube of dimension d, is at most b, with probability tending to 1 as d tends to infinity (note that the size of Q d is 2 d and the degree of Q d is d). Galvin [4] later calculated b = 5.
Kahn's result raises a new question: What happens to the range of a random homomorphism, if the degree is logarithmic, but the diameter is large? (The discrete cube has logarithmic degree, but also has logarithmic diameter.) To answer this question, we study the graph C n,k in Section 3 (the graph C n,k is the tensor product of the n-cycle and the complete graph of size k with self-loops). We show a sharp transition in k, of the range of a random homomorphism of C n,k . Namely, for any monotone function ψ(n) tending to infinity, if k = 2 log(n) − ψ(n) the range is 2 Ω(ψ(n)) , with high probability, and if k = 2 log(n) + ψ(n) the range is 3, with high probability. In particular, C n,3 log n is a graph of almost linear diameter and logarithmic degree such that the range of a random homomorphism of C n,3 log n is 3, with high probability.
The rest of this paper is organized as follows: We first introduce some notation. Section 2 contains our lower bound. Section 3 proves the upper and lower bounds on the range of random homomorphisms of the graph C n,k . Section 4 lists some further possible research directions concerning random homomorphisms of graphs.

Acknowledgement.
We would like to thank Ori Gurel-Gurevich for useful discussions.

Notation and Definitions
Logarithms are always of base 2. For an integer k ∈ N, denote [k] = {1, . . . , k}. For two integers x, y ∈ Z, denote by [x, y] the set of integers at least x and at most y. For n ∈ N, denote by Z n the additive group whose elements are [0, n − 1], and addition is modulo n.

Graphs
All graphs considered are simple and connected. Let G be a graph. For simplicity of notation, we use G to denote the vertex set of the graph G. In particular, we write v ∈ G, if v is a vertex of the graph G. For two vertices v, u ∈ G, we write v ∼ u ∈ G, if {u, v} is an edge in the graph G. When the graph is clear, we use v ∼ u. The size of the graph G, denoted |G|, is the number of vertices in G. The diameter of G is the maximal distance between any two vertices in G. For a vertex v ∈ G and an integer r ∈ N, a ball of radius r centered at v is the subgraph of G induced by the set of all vertices at distance at most r from v.

Homomorphisms
For two graphs G and H, a graph homomorphism from G to H is a mapping f : G → H that preserves edges; that is, every two vertices We denote by Z both the set of integers, and the graph whose vertex set is the integers and edge set is {z, z + 1} z ∈ Z . We mostly consider Hom v 0 (G, Z) def = Hom 0 v 0 (G, Z). Note that For a mapping f : We call both f (G) and R(f ) the range of f . We use the notation ∈ R to denote an element chosen uniformly at random. E.g., f ∈ R Hom v 0 (G, Z) is a random homomorphism from G to Z such that f (v 0 ) = 0, chosen uniformly at random. (When G is finite and connected, the set Hom v 0 (G, Z) is finite, and f ∈ R Hom v 0 (G, Z) is well defined.) For example, consider the case where G is the interval of length n; that is Then, Hom 0 (G, Z) is the set of all paths in Z starting from 0, of length n. Therefore, f ∈ R Hom 0 (G, Z) is a n-step random walk on Z, starting at 0. Thus, for a general (connected and finite) graph G, a random homomorphism f ∈ R Hom v 0 (G, Z), is also called a G-indexed random walk.
For a graph G, we say that a homomorphism f from G to itself is an automorphism, if f is invertible, and f −1 is a homomorphism as well. We say that a graph G is vertex transitive, if all the vertices of G "look" the same; that is, for any two vertices v, u ∈ G, there exists an We say that a graph G is edge transitive, if all the edges of G "look" the same; that is, for any two edges {v 1 , v 2 } and {u 1 , u 2 } in G, there exists

Lower Bounds for Graphs with Small Degree
In this section we show that for graphs of 'small enough' degree, the range of a homomorphism is typically 'large'. In fact, we prove something slightly stronger: Theorem 2.1. Let {G n } be a family of graphs such that lim n→∞ |G n | = ∞. For r ∈ N, define V n (r) to be the maximal size of a ball of radius r in G n . Let v n ∈ G n and let f n ∈ R Hom vn (G n , Z) be a random homomorphism. Let r = r(n) ∈ N. Assume that there exists a constant c < 1 such that every large enough n ∈ N admits V n (r) ≤ c log |G n |. Then We defer the proof of Theorem 2.1 to Section 2.4. First we discuss the tightness of Theorem 2.1. In Section 3 we consider the family of graphs {C n,k }, where n ∈ N is even, and k = k(n) ∈ N. For n ∈ N, the size of C n,k is kn, and the size of a ball of radius 3 in C n,k is at most 7k; that is, V n (3) ≤ 7k. In Theorem 3.1 we show an upper bound on the range of a random homomorphism of C n,k , for logarithmic k. More specifically, we show that for k = 2 log n + log log log n, where f n ∈ R Hom (0,1) (C n,k , Z) is a random homomorphism.
where f n ∈ R Hom (0,1) (C n,k , Z) is a random homomorphism. This is a contradiction to (1).
We note that since C n,k is vertex transitive (and edge transitive), Theorem 2.1 is tight in the above sense for vertex transitive graphs (and for edge transitive graphs).

Lower Bounds for Graphs with Small Degree
The following corollary of Theorem 2.1 shows that the range of a random homomorphism from a graph of "small" degree to Z is "large". For example, consider any family of graphs {G n }, such that the degree of G n is log log |G n |. Then, the corollary states that the range of a random homomorphism from G n to Z is super-constant (as n tends to infinity), with high probability.
Corollary 2.2. Let {G n } be a family of graphs such that lim n→∞ |G n | = ∞. Let n ∈ N, and let d = d(n) be the maximal degree of G n . Let v n ∈ G n and let f n ∈ R Hom vn (G n , Z) be a random homomorphism. Then Proof. For r ∈ N, denote by V n (r) the maximal size of a ball of radius r in G n . Since the maximal degree of G n is d = d(n), every r ∈ N admits V n (r) ≤ (d + 1) r . Denote r = r(n) = log log|Gn|−1 log(d+1) . Since (d + 1) r = 1 2 log |G n |, the corollary follows, by Theorem 2.1 (with c = 1/2). ⊓ ⊔

An Example -the Torus
A specific example for the use Theorem 2.1 is in the case of the torus. For an integer n ∈ N, define the n × n torus, denoted T n , as follows: The vertex set is Z n × Z n , and the edge set is defined by the relations (where addition is modulo n). Note that T n is both vertex transitive and edge transitive. The following corollary shows that the range of a random homomorphism of the n × n torus is at least Ω(log 1/2 n), with high probability. Proof. Note that the size of T n is n 2 . For r ∈ N, denote by V n (r) the maximal size of a ball of radius r in T n . The vertex set of the ball of radius r centered at (0, 0) is contained in Thus, since T n is vertex transitive, V n (r) ≤ (2r + 1) 2 . So, since every large enough n admits (2(1/2 log 1/2 n) + 1) 2 ≤ 2/3 log(n 2 ), using Theorem 2.1 (with r = 1/2 log 1/2 n and c = 2/3), For a graph G and a vertex v ∈ G, we denote by B r (v) the ball of radius r centered at v. We say that B r (v) is of exact radius r, if there exists u ∈ B r (v) such that the distance between v and u in G is at least r. and Γ = u ∈ B u ∈ B r−1 (the boundary of B). Let f be a homomorphism from B to Z.
Assume that B is of exact radius r. Then there exists a homomorphism g from B to Z such that g| Γ = f | Γ , and R(g) ≥ r + 1.
Proof. Since a translation of a homomorphism is a homomorphism with the same range size, assume without loss of generality that min u∈Γ f (u) = 0. We demonstrate an iterative process such that at the i th step (for i = 0, . . . , r) we have a homomorphism g i that admits 2. The minimal value of g i on the ball B r−i is i.
For the first step, we define g 0 = |f |; that is, for u ∈ B, define g 0 (u) = |f (u)|. Note that g 0 is a homomorphism from B to Z. Since min u∈Γ f (u) = 0, it follows that g 0 | Γ = f | Γ . Furthermore, every u ∈ B admits g 0 (u) ≥ 0. Thus, g 0 has the two properties described above.
At the i th step (i > 0), define g i as follows , and S = V n (r). We recall the following definitions. For v ∈ G, we denote by B r (v) the ball of radius r centered at v. We say that such that the distance between v and u in G is at least r. The following claim describes the size of a collection of pairwise disjoint balls of exact radius r in G.
For all u ∈ U, the ball B r (u) is of exact radius r.

For all
Proof. We prove the claim by induction on the size of V . Induction base: If |V | < S 2 , then there is nothing to prove. Induction step: Assume |V | ≥ S 2 . If r = 0, then set U = V , and the claim follows. If r > 0, then S > 1, which implies |V | > S. Then, since S is the size of Returning to the proof of Theorem 2.1: By Claim 2.5, let k = ⌊|G|/S 2 ⌋−1, and let B 1 , . . . , B k be a collection of pairwise disjoint balls of exact radius r in G such that for every i ∈ [k], we Since there are at most Since E i,g = ∅, since B i is of exact radius r, and since v 0 ∈ B i , using Lemma 2.4, there exists Note that for Since (2) holds for any g such that E i,g = ∅, where the sum is over all homomorphisms g ∈ where the last equality holds, since lim n→∞ |G n | = ∞. ⊓ ⊔

The Cycle -C n,k
In this section we study the range of a random homomorphism of the graph C n,k , where n, k ∈ N, and n is even. We consider the graph C n,k mainly for two reasons: First, for logarithmic k, the graph C n,k has almost linear diameter (the diameter of C n,k is Ω(n), while the size of C n,k is O(n log n)), and still the range of a random homomorphism of C n,k is constant (the range is 3). Second, C n,k is both vertex transitive and edge transitive.
The graph C n,k is a cycle of n layers. Each layer has k vertices, and is connected to both its adjacent layers by a complete bi-partite graph. Thus, the degree of C n,k is 2k. Formally, the vertex set of C n,k is Z n × [k], and the edge set of C n,k is defined by the relations where addition is modulo n. (C n,k is also the tensor product of the n-cycle and the complete graph on k vertices with self-loops.) Denote by H n,k = Hom (0,1) (C n,k , Z), the set of homomorphisms from C n,k to Z that map (0, 1) to 0. Since n is even, C n,k is bi-partite, which implies that H n,k = ∅.
We show a threshold phenomena (with respect to k) concerning the range of a random homomorphism from C n,k to Z. More precisely, for k(n) = 2 log n + ω(1), the range of a random homomorphism is at most 3, with high probability, and on the other hand for k(n) = 2 log n − ω(1), the range of a random homomorphism is super-constant, with high probability. The following two theorems make the above statements precise.
Theorem 3.1. Let n ∈ N be even, and let k = k(n) = 2 log n + ψ(n), where ψ : N → R + is such that lim n→∞ ψ(n) = ∞. Let f n ∈ R H n,k be a random homomorphism. Then Theorem 3.2. Let n ∈ N be even, and let k = k(n) = 2 log n − ψ(n), where ψ : N → R + is monotone and lim n→∞ ψ(n) = ∞. Let f n ∈ R H n,k be a random homomorphism. Then Pr For the rest of this section we prove the above theorems. The proof of Theorem 3.1 is deferred to Section 3.4, and the proof of Theorem 3.2 is deferred to Section 3.6. We note that for k = 1, we have that C n,k is the n-cycle. Thus, f n ∈ R H n,1 is a random walk bridge of length n (a random walk bridge is a random walk conditioned on returning to 0). In this case, Theorem 3.2 gives the bound This is consistent with the range of a random walk bridge (see also Remark 3.17).

Definitions
Recall that H n,k = Hom (0,1) (C n,k , Z) is the set of homomorphisms from C n,k to Z that map (0, 1) to 0. Denote by H 0 n,k the set of homomorphisms from C n,k to Z that map the 0-layer to 0; that is, When n and k are clear we use H 0 = H 0 n,k . For f ∈ H n,k and i ∈ Z n , we say that the i-layer is constant in f , if f gets the same value on to be the set of homomorphisms in H 0 that have exactly ℓ non-constant layers.
Loosely speaking, a homomorphism of C n,k corresponds to a path on Z that starts at 0 and ends at 0. This motivates the following definition. For an even integer m ∈ N, denote by P(m) the set of paths of length m on Z that start at 0 and end at 0; that is, Note that |P(m)| = m m/2 . Consider the values of a homomorphism on the 1-layer. Since all vertices in the 1-layer are connected to a vertex that is mapped to 0, the value of a homomorphism on the 1-layer corresponds to a vector in {1, −1} k . In fact, it turns out that the value of a homomorphism on a non-constant layer corresponds to a {1, −1} k non-constant vector. Thus, the following definition will be useful. Define

The Constant Layers
In this section we show some properties of the constant layers. First, we show that homomorphisms in H n,k do not have two adjacent non-constant layers. Second, we show that if the 0-layer is non-constant in a homomorphism f ∈ H n,k , then we can think of f as a homomorphism in H 0 n−2,k . Third, we show that, conditioned on a specific set of ℓ non-constant layers, a random homomorphism in H 0 n,k corresponds to a random walk bridge of length n − 2ℓ (i.e., a random walk of length n − 2ℓ on Z that starts at 0 and ends at 0).

No Two Adjacent Non-constant Layers
Claim 3.3. Let f ∈ H n,k be a homomorphism. Assume that i ∈ Z n is such that the i-layer is non-constant in f . Then there exists z ∈ Z such that In particular, both the (i + 1)-layer and the (i − 1)-layer are constant in f .
Proof. We prove the claim for the (i + 1)-layer. The proof for the (i − 1)-layer is similar. Let f ∈ H n,k \ H 0 n,k be a homomorphism. That is, the 0-layer is non-constant in f . Define f ↓ as follows: Proof. Since the 0-layer is non-constant in f , by Claim 3.3, there exists z ∈ {1, −1} such that First, we show that f ↓ is a homomorphism of C n−2,k . Indeed, for all i ∈ [0, n − 4] and for all (4), we have In fact, the following claim holds.
First, the mapping is injective. Indeed, let f 1 = f 2 be two homomorphisms in H n,k \ H 0 n,k . If f 1 (1, 1) = f 2 (1, 1), then the images of f 1 and f 2 are different (in the second coordinate).
Otherwise, assume that There exist i ∈ Z n and s ∈ [k] such that f 1 (i, s) = f 2 (i, s). If either i = 1 or i = n−1, since the 0-layer is non-constant in both f 1 and f 2 , using Claim 3.3, then for s > 1 (since f 1 (0, 1) = f 2 (0, 1) = 0), implying that the images of f 1 and f 2 are different. 1, s), so the images of f 1 and f 2 are different (in the first coordinate).

Conditioned on the Set of Non-constant Layers, a Random Homomorphism is a Random Walk Bridge
Let be a set of size ℓ such that for every i ∈ [n − 2], either i ∈ I or i + 1 ∈ I (or both). We think of I as a set of non-constant layers (recall Claim 3.3).
Denote by H(I, n) the set of homomorphisms f in H 0 n,k such that I is the set of non-constant layers in f (we think of k as fixed). Recall that P(n − 2ℓ) is the set of paths on Z of length n − 2ℓ that start at 0 and end at 0, and recall that For a homomorphism f ∈ H n,k , define the range of the constant layers in f to be For a path (S 0 , S 1 , . . . , S n−2ℓ ) in P (n − 2ℓ), define the range of the path to be Loosely speaking, the following proposition shows that, conditioned on the set of non-constant layers, a random homomorphism in H 0 is a random walk bridge. Proposition 3.6. Let I = {i 1 < i 2 < · · · < i ℓ } ⊆ [n − 1] be a set of size ℓ such that for all i ∈ [n − 2], either i ∈ I or i + 1 ∈ I. Then there exists a bijection ϕ between H(I, n) and P (n − 2ℓ) × V ℓ . Furthermore, denote ϕ = (ϕ 1 , ϕ 2 ). Then for all f ∈ H(I, n), For the rest of this section we prove Proposition 3.6.

Proof of Proposition 3.6
We prove the proposition by induction on ℓ. The induction step is based on the following claim. The claim shows that given a non-constant layer in a homomorphism f of C n,k , we can think of f as a homomorphism of C n−2,k .
In what follows, for simplicity, we use the following convention: For a homomorphism f ∈ H n,k , and integers i ∈ N and s ∈ [k], we define f (i, s) = f (i (mod n), s). Claim 3.7. Let I = {i 1 < · · · < i ℓ } ⊆ [n − 1] be a set of size ℓ such that for every i ∈ [n − 2], either i ∈ I or i + 1 ∈ I. Let f ∈ H(I, n) be a homomorphism. Define Then the map Φ : Proof. First we show that f ′ and v f are well defined: We are left with the case i = i ℓ − 1. Since the i ℓ -layer is non-constant in f , by Claim 3.3, ). Thus, is constant in f , which implies that the i-layer is constant in f ′ . So, the set of non-constant layers in f ′ is the set I \ {i ℓ }, and f ′ ∈ H(I \ {i ℓ } , n − 2). Now we show that v f is well defined: Since the i ℓ -layer is non-constant in f , using Claim 3.3, To show that Φ is a bijection, we provide the inverse map Ψ = Φ −1 . Define as follows: For a pair g ′ ∈ H(I \ {i ℓ } , n − 2) and v ∈ V , define g ∈ H(I, n) by and define Ψ(g ′ , v) = g.
Proof. Choose some i ∈ [0, n − 1] and s, t ∈ [k]. If i + 1 < i ℓ , then If i ≥ i ℓ + 1, then If i = i ℓ − 1, then, using Claim 3.3, If i = i ℓ , then, using Claim 3.3, ). Thus, for i < i ℓ , we have that the i-layer is constant in g ⇔ the i-layer is constant in g ′ .

So we have that the
Thus, the set of non-constant layers of g is Thus, for all i ∈ [0, n − 1] and s ∈ [k], So g ≡ f , and Ψ = Φ −1 . Thus, Φ is a bijection as claimed.
We can exclude the case i = i ℓ , because the i ℓ -layer is non-constant in f .
If i = i ℓ + 1, then using Claim 3.3, for all s ∈ [k], since the i-layer is constant in f , So, the (i + 2)-layer is constant in f and Back to the proof of Proposition 3.6. Loosely speaking, we define ϕ to be ℓ compositions of Φ, where Φ is the map defined in Claim 3.7.

The Size of H 0
Fix n, k ∈ N (n is even) for the rest of this section. We consider H 0 = H 0 n,k . Note that, by Claim 3.3, every f ∈ H 0 has at most n/2 non-constant layers. Thus, where H 0 (ℓ) is the set of homomorphisms of C n,k that have exactly ℓ non-constant layers.
The following lemma gives a formula for the size of H 0 (ℓ).
Proof. If k = 1, then proving the lemma. So assume that k > 1. Define a family of sets I = I ⊆ [n − 1] |I| = ℓ and for every i ∈ [n − 2], either i ∈ I or i + 1 ∈ I .
Let f ∈ H 0 (ℓ) be a homomorphism. Recall that N C(f ) is the set of non-constant layers of f .

⊓ ⊔
Back to the proof of Lemma 3.9. By (9), where H(I, n) is the set of homomorphisms f ∈ H 0 such that N C(f ) = I. By Proposition 3.6, for every I ∈ I, By Claim 3.10, So the lemma follows. ⊓ ⊔

Upper
Bound for k = 2 log n + ω(1) -Theorem 3.1 In this part we show that for k = 2 log n + ω(1), the range of a random homomorphism from C n,k to Z is 3, with high probability. We use the formula for |H 0 (ℓ)| to conclude that f has n/2 non-constant layers, with high probability. Which implies that f is "almost" constant.

Many Non-constant Layers
To prove Theorem 3.1, we use the following lemma, which states that there are n/2 nonconstant layers in a random homomorphism of C n,k . Note that, by Claim 3.3, the maximal number of non-constant layers in every homomorphism of C n,k is n/2.

Proof of Theorem 3.1
Denote f = f n , and consider the following two cases: Case one: Assume that f ∈ R H 0 n,k . By Lemma 3.11, with probability 1 − o(1), there are n/2 non-constant layers in f . Thus, since n is even and since the 0-layer is constant in f , using

The Number of Non-constant Layers Determines the Range
In the previous section we have seen that if the number of non-constant layers is large, then the range is small. In this section we show that, in fact, the number of non-constant layers determines the range of a random homomorphism. The following lemma gives a lower bound on the range of a random homomorphism of C n,k with exactly ℓ non-constant layers. The lower bound is determined by ℓ. We note that a similar upper bound can be proven.
Lemma 3.12. Let n ∈ N be even, and let k = k(n) ∈ N. Let ℓ = ℓ(n) ∈ N be such that lim n→∞ (n − 2ℓ) = ∞. Let f n ∈ R H 0 n,k (ℓ) be a random homomorphism from C n,k to Z with exactly ℓ non-constant layers such that f ({0} × [k]) = {0}. Then for every α > 0, where the o(1) term is as n tends to infinity, and is independent of α.
Loosely speaking, the proof of the lemma is as follows. Conditioned on f having ℓ nonconstant layers, f corresponds to a random walk bridge of length n − 2ℓ. Since the range of such a walk is roughly √ n − 2ℓ, the range of f is roughly √ n − 2ℓ.

Proof of Lemma 3.12
Recall that for an even m ∈ N, we defined P(m) to be the set of paths on Z of length m that start at 0 and end at 0. The following proposition shows that, with high probability, the range of a random walk bridge of length m is at least Ω( √ m).
Proposition 3.13. Let m ∈ N be even, and let (S 0 , S 1 , . . . , S m ) ∈ R P(m) be a random walk bridge of length m. Then for every α > 0, where the o(1) term is as m tends to infinity, and is independent of α.
Let f = f n ∈ R H 0 (ℓ) be a random homomorphism. Then, for all I such that H(I, n) = ∅, where (S 0 , , . . . , S n−2ℓ ) ∈ R P(n − 2ℓ) is a random walk bridge. Thus, we have where the sum is over all sets I ⊆ [n − 1] such that H(I, n) = ∅. Thus, by Proposition 3.13, since lim n→∞ (n − 2ℓ) = ∞, and the o(1) term is as n tends to infinity, and is independent of α. ⊓ ⊔
Let T ∈ [m]. Before proving the proposition we show that a path in Z of length m from 0 to 0 that passes through T corresponds to a path in Z of length m from 0 to 2T . Formally, Claim 3.14. There exists a bijection between paths in P(m) that pass through T and paths in Z of length m that start at 0 and end at 2T .
Proof. Let (S 0 , S 1 , . . . , S m ) ∈ P(m) be such that there exists j ∈ [m] that admits S j = T . Let The bijection is reflecting the path around T from j * onwards.
That is, for j ∈ [0, j * ] set S ′ j = S j , and for j ∈ [j * + 1, m] set S ′ j = 2T − S j . Thus, S ′ 0 = 0, S ′ j * = T and S ′ m = 2T . Furthermore, (S ′ 0 , S ′ 1 , . . . , S ′ m ) is a path in Z of length m such that S ′ 0 = 0 and S ′ m = 2T . Note that j * is also the first time that S ′ passes through T . Since every path in Z of length m that starts at 0 and ends at 2T passes through T , the above defined map is a bijection. Indeed, we show how to invert the above defined map. Let Since there are m m/2−T paths in Z of length m that start at 0 and end at 2T , using Claim 3.14, Using Stirling's formula (recall that for x ≥ 0, we have 1 − x ≤ e −x and 1 + x ≤ e x ), where the o(1) term is as m tends to infinity, and is independent of α, since α < 1.
⊓ ⊔ 3.6 A Lower Bound for k = 2 log n − ω(1) -Theorem 3.2 In this part we show that for k = 2 log n − ω(1), the range of a random homomorphism from C n,k to Z is super-constant, with high probability. The proof plan is as follows: First, we prove that there are many constant layers in a random homomorphism f . Second, using Lemma 3.12, we conclude that the range of f is large.

Many Constant Layers
The following lemma shows that a random homomorphism of C n,k has many constant layers.

Proof of Theorem 3.2
First we consider homomorphisms in H 0 .
Case two: Assume that f ∈ R H n,k \ H 0 n,k . By Claim 3.5, we have that f ↓ is uniformly distributed in H 0 n−2,k . Thus, by Claim 3.16, with probability at least the range of f ↓ is at least ψ(n − 2) −1 2 ψ(n−2)/4 .
By definition of f ↓ , the size of the range of f is at least the size of the range of f ↓ . So, with probability 1 − o(1), the range of f is at least ψ(n) −1 2 ψ(n−2)/4 (recall that ψ(n) is monotone).

Further Research
We list some possible further research directions regarding random homomorphisms of graphs: Note that a homomorphism is always Lipschitz, but typically the set of homomorphisms is much smaller than the set of Lipschitz functions. For a graph G and a vertex v ∈ G, define Lip 0 v (G, Z) to be the set of all Lipschitz functions from G to Z that map v to 0.
Conjecture. Let {G n } be a family of bi-partite graphs with lim n→∞ |G n | = ∞. Assume that for all n, G n has maximal degree d (d independent of n). Let f n ∈ R Hom 0 vn (G n , Z) be a random homomorphism, and let g n ∈ R Lip 0 vn (G n , Z) be a random Lipschitz function. Then, where Θ(·) depends on d.
2. Let G be a bi-partite graph, and let ∆ be the diameter of G. For any homomorphism f ∈ Hom 0 v (G, Z), we have that R(f ) = O(∆). But this naive bound should not be the typical bound, at least not for symmetric graphs.